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eHANDBOOK
Level Measurement
Spring 2019
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www.controlglobal.com
eHandbook: Level Measurement, Spring 2019 2
TABLE OF CONTENTSWhen is reducing variability wrong? 4
Level may vary the most when process and product variation are minimized.
Solutions to prevent harmful feedforwards 8
How to correct issues in boiler, distillation column and neutralization control.
Understanding P, I and D 12
The simple mathematics can be clarified with mechanical analogies and an example of level control.
Is global warming like level control? 20
Comparing the processes sheds light on the problem of regulating Earth’s temperature.
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eHandbook: Level Measurement, Spring 2019 4
When is reducing variability wrong?Level may vary the most when product and process variation are minimized.by Greg McMillan
Having the blind wholesale goal of reducing variability can lead to doing the wrong thing that can reduce plant safety and performance. Here
we look at some common mistakes made
that users may not realize until they have
better concepts of what is really going on.
We seek to provide some insightful knowl-
edge here to keep you out of trouble.
Is a smoother data historian plot or a statis-
tical analysis showing less short-term vari-
ability good or bad? The answer is no for
the following situations, misleading users
and data analytics.
First of all, the most obvious case is surge
tank level control. Here we want to maxi-
mize the variation in level to minimize the
variation in manipulated flow, typically to
downstream users. This objective has a
positive name of absorption of variability.
What this is really indicative of is the prin-
ciple that control loops don’t make vari-
ability disappear, but transfer variability
from a controlled variable to a manipulated
variable. Process engineers often have a
problem with this concept because they
think of setting flows per a Process Flow
Diagram (PFD) and are reluctant to let a
controller freely move them per some al-
gorithm they don’t fully understand. This is
seen in predetermined sequential additions
of feeds or heating and cooling in a batch
operation rather allowing a concentration or
temperature controller do what is needed
via fed-batch control. No matter how smart
a process engineer is, not all of the situa-
tions, unknowns and disturbances can be
accounted for continuously. This is why fed-
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eHandbook: Level Measurement, Spring 2019 5
batch control is called semi-continuous. I
have seen where process engineers, believe
or not, sequence air flows and reagent flows
to a batch bioreactor rather than going to
Dissolved Oxygen or pH control. We need
to teach chemical and biochemical engi-
neers process control fundamentals, includ-
ing the transfer of variability.
The variability of a controlled variable is
minimized by maximizing the transfer of
variability to the manipulated variable. Un-
necessary sharp movements of the ma-
nipulated variability can be prevented by
a setpoint rate of change limit on analog
output blocks for valve positioners or VFDs,
or directly on other secondary controllers
(e.g., flow or coolant temperature), and the
use of external-reset feedback (e.g., dy-
namic reset limit) with fast feedback of the
actual manipulated variable (e.g., position,
speed, flow or coolant temperature). There
is no need to re-tune the primary process
variable controller by the use of external-
reset feedback.
Data analytics programs need to use ma-
nipulated variables in addition to controlled
variables to indicate what is happening.
For tight control and infrequent setpoint
changes to a process controller, what is
really happening is seen in the manipulated
variable (e.g., analog output).
A frequent problem is data compression in
a data historian that conceals what is really
going on. Hopefully, this is only affecting
the trend displays and not the actual vari-
ables being used by a controller.
The next most common problem has been
extensively discussed by me, so at this point
you may want to move on to more pressing
needs. This problem is the excessive use of
signal filters that may even be more insidi-
ous because the controller doesn’t see a de-
veloping problem as quickly. A signal filter
that is less than the largest time constant in
the loop (hopefully in the process) creates
dead time. If the signal filter becomes the
largest time constant in the loop, the previ-
ously largest time constant creates dead
time. Since the controller tuning based on
largest time constant has no idea where
it is, the controller gain can be increased,
which, combined with the smoother trends,
We need to teach chemical and biochemical
engineers process control fundamentals,
including the transfer of variability.
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eHandbook: Level Measurement, Spring 2019 6
can lead one to believe the large filter was
beneficial. The key here is a noticeable in-
crease in the oscillation period, particularly
if the reset time was not increased. Signal
filters become increasingly detrimental as
the process loses self-regulation. Integrat-
ing processes such as level, gas pressure
and batch temperature are particularly
sensitive. Extremely dangerous is the use of
a large filter on the temperature measure-
ment for a highly exothermic reaction. If the
PID gain window (ratio of maximum to mini-
mum PID gain) reduces due to measure-
ment lag to the point of not being able to
withstand nonlinearities (e.g., ratio less than
6), there is a significant safety risk.
A slow thermowell response, often due to
a sensor that is loose or not touching the
bottom of the thermowell, causes the same
problem as a signal filter. An electrode that
is old or coated can have a time constant
that is orders of magnitude larger (e.g., 300
sec) than a clean, new pH electrode. If the
velocity is slightly low (e.g., less than 5 fps),
pH electrodes become more likely to foul
and if the velocity is very low (e.g., less than
0.5 fps), the electrode time constant can
increase by one order of magnitude (e.g.,
30 sec) compared to an electrode seeing
recommended velocity. If the thermowell
or electrode is being hidden by a baffle, the
response is smoother but not representa-
tive of what is actually going on.
For gas pressure control, any measure-
ment filter, including that due to transmitter
damping, generally needs to be less than
0.2 sec, particularly if volume boosters on
valve positioner output(s) or a variable-
frequency drive is needed for a faster
response.
Practitioners experienced in doing Model
Predictive Control (MPC) want data com-
pression and signal filters to be completely
removed so that the noise can be seen and
a better identification of process dynamics,
especially dead time, is possible.
Virtual plants can show how fast the ac-
tual process variables should be changing,
revealing poor analyzer or sensor resolution
and response time, and excessive filtering.
In general, you want measurement lags to
total up to less than 10% of the total loop
dead time, or less than 5% of reset time.
However, you can’t get a good idea of the
loop dead time unless you remove the
filter and look for the time it takes to see a
change in the right direction, beyond noise,
after a controller setpoint or output change.
For more on the deception caused by a
measurement time constant, see the Con-
trol Talk Blog, “Measurement Attenuation
and Deception.”
http://www.controlglobal.com/blogs/controltalkblog/measurement-attenuation-and-deception-tipshttp://www.controlglobal.com/blogs/controltalkblog/measurement-attenuation-and-deception-tips
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eHandbook: Level Measurement, Spring 2019 8
Solutions to prevent harmful feedforwardsHow to correct issues in boiler, distillation column and neutralization control.by Greg McMillan
Here we look at applications where feedforward can do more harm than good, and what to do to prevent this situation. This problem is
more common than one might think. In the
literature, we mostly hear how beneficial
feedforward can be for measured load
disturbances. Statements are made that
the limitation is the accuracy of the feed-
forward and that, consequently, an error of
2% can still result in a 50:1 improvement in
control. This optimistic view doesn’t take
into account process, load and valve dy-
namics. The feedforward correction needs
to arrive in the process at the same point
and the same time as the load disturbance.
This is traditionally achieved by passing
the feedforward (FF) through a deadtime
block and lead-lag block. The FF dead-
time is set equal to the load path deadtime
minus the correction path deadtime. The
FF lead time is set equal to the correction
path lag time. The FF lag time is set equal
to the load path lag time. If the FF arrives
too soon, we create inverse response,
and if the FF arrives too late, we create
a second disturbance. Setting up tuning
software to identify and compute the FF
dynamic can be challenging. Even more
problematic are the following feedforward
applications that do more harm than good
despite dynamic compensation.
1. Inverse response from the manipulated
flow causes excessive reaction in the
opposite direction of load. The inverse
response from a feedwater change can be
so large as to cause a boiler drum high or
low level trip, a situation that particularly
occurs for undersized drums and miss-
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eHandbook: Level Measurement, Spring 2019 9
ing feedwater heaters due to misguided
attempts to save on capital costs. The
solution here is to use a traditional three-
element drum level control, but added to
the traditional feedforward is an uncon-
ventional feedforward with the opposite
sign that is decayed out over the period of
the inverse response. In other words, for a
step increase in steam flow, there would be
initially a step decrease in boiler feedwater
feedforward added to the three-element
drum level controller output that is trying
to increase feedwater flow. This prevents
shrink and a low level trip from bubbles
collapsing in the downcomers from an
increase in cold feedwater. For a step de-
crease in steam flow, there would be a step
increase in boiler feedwater feedforward
added to the three-element drum level
controller output that is trying to decrease
feedwater flow. This prevents swell and
a high level trip from bubbles forming in
the downcomers from a decrease in cold
feedwater. A severe problem of inverse re-
sponse can occur in furnace pressure con-
trol when the scale is a few inches of water
column and the incoming manipultaed
air is not sufficiently heated. The inverse
response from the ideal gas law can cause
a pressure trip. An increase in cold air flow
causes a decrease in gas temperature and,
consequently, a relatively large decrease
in gas pressure at the furnace pressure
sensor. A decrease in cold air flow causes
an increase in gas temperature and, con-
sequently, a relatively large increase in gas
pressure at the furnace pressure sensor.
2. Deadtime in the correction path is
greater than deadtime in the load path. The
result is a feedforward that arrives too late,
creating a second disturbance and worse
control than if there was no feedforward.
This occurs whenever the correction path
is longer than the load path. An example is
a distillation column control when the feed
load upset stream is closer to the tempera-
ture control tray than the corrective change
in reflux flow. The solution is to generate the
feedforward signal for ratio control based
on a setpoint change that is then delayed
before being used by the feed flow control-
ler. The delay is equal to the correction path
deadtime minus the load path deadtime. The
The result is a feedforward that arrives too late,
creating a second disturbance and worse control
than if there was no feedforward.
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eHandbook: Level Measurement, Spring 2019 10
same problem can occur for a reagent injec-
tion delay that often occurs due to conven-
tionally-sized dip tubes and small reagent
flows. The same solution applies in terms of
using an influent flow controller setpoint for
feedforward ratio control of reagent, and
delaying the setpoint used by the influent
flow controller.
3. Feedforward correction makes re-
sponse from an unmeasured disturbance
worse. This occurs in unit operations
such as distillation columns and neutral-
izers where the unmeasured disturbance
from a feed composition change is made
worse by a feedforward correction based
on feed flow. Often, feed composition is
not measured and is large due to parallel
unit operations and a combination of flows
that become the feed flow. For pH, the
nonlinearity of titration curves increases
the sensitivity to feed composition. Even if
the influent pH is measured, the pH elec-
trode error or uncertainty of the titration
curve makes feedforward correction for
feed pH to do more harm than good for
setpoints on the steep part of the curve.
If the feed composition change requires
a decrease in manipulated flow and there
is a coincidental increase in feed flow that
corresponds to an increase in manipulated
flow or vice-versa, the feedforward does
more harm than good. The solution is to
compute the required rate of change of
manipulated flow from the unmeasured
disturbance, and add this to the computed
rate of change for the feedforward correc-
tion needed, paying attention to the signs
of the rate of change. If the required rate
of change of manipulated flow for the un-
measured disturbance is in the opposite di-
rection, the feedforward correction rate of
change in manipulated flow is decreased. If
it exceeds the computed feedforward cor-
rection rate of change in the manipulated
flow, the feedforward rate of change is
clamped at zero to prevent making con-
trol terribly worse. If the required rates of
change for the manipulated flow are in the
same direction, the magnitude of the feed-
forward rate of change is correspondingly
increased.
I am trying to see how all this applies in my
responses to known and unknown upsets to
my spouse.
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eHandbook: Level Measurement, Spring 2019 12
Understanding P, I and DThe simple mathematics can be clarified with mechanical analogies and an example of level control.by R. Russell Rhinehart
It is important to understand what the proportional, integral and derivative terms do within the PID controller. That understanding is essential to choose ap-
propriate action, troubleshoot controllers,
choose appropriate modifications, or set
up advanced controllers. Unfortunately, the
controller synthesis approach, in which PID
magically appears within Laplace Transform
analysis, does not provide that functional
understanding. Hopefully, this more intuitive
development of PID will be helpful.
Chosen as an example is a commonly un-
derstood process—level control in a tank of
liquid (Figure 1). The inflow is a wild variable,
or disturbance, that will upset the level, which
is an indication of liquid inventory. The slide
valve on the outflow will open or close to re-
lease more or less fluid to keep the level at
the desired setpoint. As notation, the con-
trolled variable, CV, is the liquid level in the
tank, h, and the manipulated variable, MV, is
the valve stem position, U. Recognize that
your region or community may use alternate
terminology. The nominal, initial steady state
values are h0 and U0.
The controller, also shown in Figure 1, is
a mechanical lever, a proportional-only
controller. If the liquid level rises somewhat,
then the float rises the same amount, which
raises the lever that opens the valve an ad-
ditional amount by the lever proportional-
ity. This releases fluid faster, which seeks
to counter the rising liquid inventory. If the
liquid level falls somewhat, the lever closes
the valve proportionally.
The liquid level may rise or fall for any
number of reasons. The inflow rate, Fin,
may change, the viscosity of the fluid may
change affecting its outflow speed, or the
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eHandbook: Level Measurement, Spring 2019 13
downstream pressure or in-pipe flow restric-
tions may change, affecting Fout. The reason
for a rise or fall in level is immaterial for the
controller. This lever action moves the valve
in the appropriate direction, and in a manner
proportional to the level change.
PROPORTIONAL ACTIONThe lever arm length ratio, b/a, is the gain
of the controller. By changing the relative
lengths of the lever arms, perhaps by chang-
ing the position of the fulcrum, the controller
can be more or less aggressive. If the level, h,
starts at a base case of h0, which is also the
setpoint, hSP, then using simple relations, the
equation for the change in the valve stem po-
sition with respect to level is:
∆U = b a ∆h = - b a (hSP - h) = Kce (1)
Where Kc = - b a represents the controller gain
and e = (hSP - h) represents the actuating er-
ror, the deviation of the CV from its setpoint.
Since ∆U = U - U0, then the equation rep-
resenting the lever-type control is:
U = U0 + Kc e (2)
Figure 2 shows a block diagram of a con-
trolled process using generic symbols for
the MV, U; the CV, Y; and the disturbance, d.
Contrasting the illustration of the physical
process of Figure 1, this represents the path
of cause-and-effect information exchange.
The controller is shown acting on the actu-
ating error, e.
Note that although termed the process out-
put, Y, the level of the liquid does not come
out of the tank. The material liquid goes in
or comes out, and level is a measure of the
inventory response of the in-tank contents.
The block diagram reveals the information
LEVEL CONTROL EXAMPLEFigure 1: Here, inflow is a variable that will upset the level gauge float, which acts on the lever arm that controls the slide valve on the outflow. The controlled variable, CV, is the liquid level in the tank, h, and the manipulated variable, MV, is the valve stem position, U.
ho
Fin
Uo
b
a
Fout
ho
Fin
Uo
Uo
U
b
a
Fout
d
U Y
U
C+
-
+
+
Pe
e
YSP
Kc
Uo Fixed bias
Turnbuckle
Oppositethreads
+
+
+
+
e
ê
Kcess
+
+Kc
∫dt1τ1
1τ1s
τps
Û
LEVEL CONTROL BLOCK DIAGRAMFigure 2: A block diagram of the control scheme in Figure 1 shows the information transfer between controller and process, not material exchange. The block labeled C is the controller, which represents the lever, but could be a calculator that executes Equation (2): it multiplies Kc times e, then adds it to U0.
ho
Fin
Uo
b
a
Fout
ho
Fin
Uo
Uo
U
b
a
Fout
d
U Y
U
C+
-
+
+
Pe
e
YSP
Kc
Uo Fixed bias
Turnbuckle
Oppositethreads
+
+
+
+
e
ê
Kcess
+
+Kc
∫dt1τ1
1τ1s
τps
Û
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eHandbook: Level Measurement, Spring 2019 14
transfer between controller and process, not
material exchange. The lines in Figure 2 are
not pipes. The block labeled C is the control-
ler, which represents the lever, but could be
a calculator that executes Equation (2). It is
simple arithmetic: the controller multiplies Kc
times e, then adds it to U0.
Equation (2) seems very much like what is
often presented as a proportional control-
ler, U = Kc e. However, if U = Kc e was the
relation, then if the CV were at the setpoint
(e = 0), the controller would set U = 0, and
close the valve, which would make the level
rise, and cause it to deviate from the set-
point. In Equation (2), the term U0 is the
controller bias—it’s the value of U for which
the initial MV position is required to hold
the CV at setpoint. As illustrated in Figure 1,
this is about 50%.
The user chooses the controller gain. Nor-
mally, the controller starts in manual mode
(MAN) with the user deciding the MV value,
then when the CV is at the setpoint, the
user switches the controller to automatic
(AUTO) mode. For bumpless transfer, the
bias is usually set by the controller as the
MV value when the controller is switched
from MAN to AUTO.
Figure 3 shows a block diagram of the arith-
metic operations in the P-only control logic
of Equation (2). The actuating error is multi-
plied by the controller gain, and then added
to the bias to determine the controller out-
put. These are simple arithmetic operations
(not calculus or Laplace-transformed magic).
Note that It does not matter whether the
controller in Figure 1 is an actual physical
float-and-lever device, or whether it’s a digi-
tal calculation of Equation (2) in a computer
that sends the valve stem position target to
an i/p device to move the valve stem. The
logic and action are identical.
STEADY STATE OFFSET AND INTEGRAL ACTIONP-only control is often good enough, but
its problem is steady state offset. Consider
what happens if Fin increases and holds at
a new value, when the level is initially at the
setpoint. Initially, Fout remains the same be-
cause h has not yet changed, and the valve
stem position is at the initial U0. Then, since
the new inflow is greater than the outflow,
level rises. As h rises, this increases U, which
increases Fout Eventually, Fout will match Fin
and the level will stop rising, but at this new
PROPORTIONAL ONLYFigure 3: A block diagram of the P-only control calculation of Equation (2) shows the actuating error is multiplied by the controller gain, and then added to the bias to deter-mine the controller output. These are simple arithmetic operations (not calculus or Laplace-transformed magic).
ho
Fin
Uo
b
a
Fout
ho
Fin
Uo
Uo
U
b
a
Fout
d
U Y
U
C+
-
+
+
Pe
e
YSP
Kc
Uo Fixed bias
Turnbuckle
Oppositethreads
+
+
+
+
e
ê
Kcess
+
+Kc
∫dt1τ1
1τ1s
τps
Û
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eHandbook: Level Measurement, Spring 2019 15
steady state, h is not at the setpoint. It must
be above the setpoint for the valve to be
open enough to let the outflow be higher.
This h deviation is steady state offset.
It is immaterial whether the disturbance is
the inflow rate or some other aspect that af-
fects either the inflow or outflow, or wheth-
er the level falls or rises as a response to the
disturbance. If the disturbance persists, the
process will not settle at the setpoint.
A method to eliminate steady state offset
is to add integral action. But, that calculus
terminology has little physical meaning, so
to add understanding, consider the injection
of a turnbuckle to the valve stem (Figure 4).
In a turnbuckle, the ends of the rods are
threaded to fit into the threaded holes of
the buckle. The threads go in opposite
directions. So, if the buckle is turned in one
direction, the two sections of the valve stem
are pulled together, the stem is shortened,
and the valve opens. If turned in the other
direction, the stem is lengthened and the
valve closes. This permits opening and clos-
ing of the valve without a change in tank
level.
If you notice that the liquid level has risen,
you know that some disturbance is acting,
and the buckle needs to be turned to short-
en the stem, to open the valve a bit more
than the original, pre-disturbance position
for the tank level. If the level rises a little bit,
there is only a small upset, and the buckle
only needs to be turned a little. In contrast,
a large level deviation indicates a large
upset has occurred, which justifies a large
turnbuckle readjustment. And, of course, a
level rise or drop would direct turns in the
opposite direction.
So, let’s have an observer follow this rule:
At each sampling, observe the level devia-
tion from setpoint, and make an incremental
change in the turnbuckle angle that’s pro-
portional to the level deviation. In Equation
(3), ∝ represents the thread pitch (axial
distance per angle), β is the proportionality
rule to change angle due to level deviation
from setpoint (angle per h deviation), and
c=∝β is their product:
∆U = ∝ ∆θ = ∝ βe = ce (3)
After the most recent sampling, the ith ob-
servation, control action changes the valve
stem length from the previous length:
INTEGRAL IS LIKE A TURNBUCKLEFigure 4. Integral action eliminates steady state offset, like adding a turnbuckle to the valve stem. If the buckle is turned in one direction, the stem is shortened and the valve opens; if turned in the other direction, the stem is lengthened and the valve closes, without a change in the tank level.
ho
Fin
Uo
b
a
Fout
ho
Fin
Uo
Uo
U
b
a
Fout
d
U Y
U
C+
-
+
+
Pe
e
YSP
Kc
Uo Fixed bias
Turnbuckle
Oppositethreads
+
+
+
+
e
ê
Kcess
+
+Kc
∫dt1τ1
1τ1s
τps
Û
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eHandbook: Level Measurement, Spring 2019 16
Ui = U i - 1 + cei (4)
The sequence of the past two adjustments is:
Ui = U i-2 + cei + ce i-1 (5)
Continuing to include past adjustments
generates an equation that indicates how all
of the adjustments have contributed to the
current valve stem length change from the
initial value, l0:
Ui = U0 + c∑Ni=1 ei (6)
In Equation (6), the number of items in the
sum is N = t ∆t , where t is the total time that
the controller has been in AUTO, and ∆t is the
controller sampling interval. Multiplying and
dividing the sum by ∆t reveals that the sum
of rectangles (e-height times ∆t-base) is just
the rectangle rule of integration, which can be
represented by the calculus notation.
U(t) = Ui = U0 + c ∆t ∑ ei ∆t = U0 +
c ∆t ∫
t o edt (7)
Note that, even though the integral symbol
is used in Equation (7), no calculus proce-
dure was used by the turnbuckle adjuster. If
this were to be implemented by a comput-
er, the Equation (4) adjustment in the valve
stem length is not calculus, but a simple
algebraic multiplication and addition. Fur-
ther, in the computer, the subscripts are not
needed. The assignment statement repre-
senting the Equation (4) action is U: = U +
ce. Don’t let the integral symbol misdirect
your understanding. There is no calculus to
the doing of control.
A common form of the controller calcula-
tion is to incrementally sum (integrate) the
scaled error, KC e, which means that the c ∆t
coefficient needs to be divided by KC. Since
the term tKCc only has dimensions of time, τi,
one can represent the PI controller function
as the block diagram of Figure 5. It shows
the integral value (really, it’s just the sum of
incremental changes) is added to the initial
bias to make the controller bias adjust at
each sampling. The block diagram nota-
tion indicates the function (inside the box)
and the function argument (the box input
value). But again, don’t be thinking calculus,
the integral operation is simply an arithme-
tic incremental accumulation.
In the standard form, KC is the controller
gain that multiplies both the P and I terms,
STANDARD PI CONTROLLERFigure 5. A block diagram of the standard form of the PI controller shows the integral value (the sum of incremental changes) is added to the initial bias to make the control-ler bias adjust at each sampling. The notation indicates the function (inside the box) and the function argument (the box input value).
ho
Fin
Uo
b
a
Fout
ho
Fin
Uo
Uo
U
b
a
Fout
d
U Y
U
C+
-
+
+
Pe
e
YSP
Kc
Uo Fixed bias
Turnbuckle
Oppositethreads
+
+
+
+
e
ê
Kcess
+
+Kc
∫dt1τ1
1τ1s
τps
Û
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eHandbook: Level Measurement, Spring 2019 17
and the integral time, τi, divides the integral
(which is actually calculated by incremental
summation).
Recall, the purpose of the proportional ac-
tion was to immediately counter the effect
of a disturbance, but its problem is that it
leaves a steady-state offset. The integral
purpose was not to be the primary control
action, but to remove the offset left by the
P-action. Accordingly, tune the P first (KC)
to set the aggressiveness of the controller,
then adjust the I-action (τi) to remove the
residual offset at a desirable rate.
ANTICIPATED ERROR AND DERIVATIVE MODEIn the description of P-action, the controller
acts on the initial impact of the disturbance
on the CV. However, the initial reveal that a
disturbance has happened might be a small
CV deviation, but, if allowed to fully develop
over time, it might evolve to a large value.
The fully developed CV deviation, not the
initial indication, represents the magnitude
of the disturbance that’s causing the CV
to start deviating. With derivative action,
the controller will take proportional action
based on the anticipated error, not just on
the initial reveal of the CV deviation. The
question is how to forecast the anticipated
error, the fully developed result of a distur-
bance?
If the process is linear and has a first-order
response to a disturbance, then the model
of how e would respond to a change in a
disturbance, ∆d, is:
τd dedt
+ e = Kd (∆d) = eanticipated (8)
The value of eanticipated is the steady, fully de-
veloped, anticipated value.
Note that if the disturbance could be mea-
sured and the process gain to the distur-
bance were known, then eanticipated could be
calculated from Kd (∆d) = eanticipated. However,
those are often unmeasured and unknown.
Fortunately, Equation (8) reveals that one
can estimate the anticipated future error
based on the current actuating error and
its rate of change, the values of which are
already known by the controller.
eanticipated = τd dedt
+ e (9)
Equation (9) does not specify what the dis-
turbance is. The deviation could indicate a
confluence of several disturbances, includ-
ing the MV. Equation (9) is called a “lead,”
which should be familiar. It represents what
a ball thrower must do to have a running
target catch the ball. The ball must be
thrown to where the receiver will be when
the ball gets there. The PI controller with
the P-action based on eanticipated is:
U = KC eanticipated + KCτi
∫ edt + U0 (10)
When Equation (9) is substituted into Equa-
tion (10) and rearranged, the PI controller
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eHandbook: Level Measurement, Spring 2019 18
with P-action on the anticipated error is the
classical PID relation:
U = KC e + KCτi
∫ edt + KC τd dedt
+ U0 (11)
Although Equation (11) looks like calculus
with its ∫ edt and dedt representation, the
integral is actually just the incrementally
updated sum, and the derivative will be cal-
culated from a numerical approach (enew - eold)∆t, which, again, is simple arithmetic subtrac-
tion and division.
PD-action is equivalent to P-action on
the anticipated error. Whether D action
is used or not, one still needs incremental
adjustment to the bias, I-action, to remove
steady-state offset.
If the process measurement is noisy, the
numerical derivative amplifies the noise im-
pact. And, if the process is relatively quick
to respond, there’s no need to use the antic-
ipated error concept. So, only use D-action
on noiseless and slow-to-evolve processes.
In block diagram notation, the PID con-
troller (a PI controller with P based on the
anticipated error, and the incremental ad-
justment to remove steady state offset) is
represented in Figure 6. The figure uses the
Laplace transformed notation, where s in-
dicates the operation to take the derivative
of the input to the function block, and 1/s
indicates to integrate the input. But, again,
regardless of the symbols or calculus words
to describe the functions all are simple
arithmetic operations.
PID IN SUMMARYProportional control is the simple concept
of taking immediate proportional action on
the actuating error, but P-only control,
U = KC e + b0, with a fixed bias, leaves
steady state offset. The user chooses the
value for KC to set controller aggressive-
ness.
Integral action incrementally adjusts the
bias to remove steady state offset. Note, al-
though called “integral,” there’s no calculus
in the action in the incremental accumula-
tion. The user chooses the value for τi to set
the speed at which offset is removed.
Derivative action forecasts what the actuat-
ing error will be, as a result of past influenc-
es, if they’re left uncontrolled. It’s the lead
commonly used in hitting a moving target.
PD-action is equivalent to P-action on the
anticipated error, and leaves steady state
offset. Note, although called “derivative,”
STANDARD PID CONTROLLERFigure 6. This block diagram of a PID control-ler uses the Laplace transformed notation, where s indicates the operation to take the derivative of the input to the function block, and 1/s indicates to integrate the input.
ho
Fin
Uo
b
a
Fout
ho
Fin
Uo
Uo
U
b
a
Fout
d
U Y
U
C+
-
+
+
Pe
e
YSP
Kc
Uo Fixed bias
Turnbuckle
Oppositethreads
+
+
+
+
e
ê
Kcess
+
+Kc
∫dt1τ1
1τ1s
τps
Û
-
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eHandbook: Level Measurement, Spring 2019 19
there is no calculus in the action of numeri-
cally estimating the CV rate of change. The
user chooses the value for τd to lead the aim
of the controller.
We communicate the PID procedure with
calculus, or Laplace or Z-transforms, or oth-
er advanced mathematical symbols. With a
bit of sarcasm, it seems the reason to use
fancy mathematics is to make people think
it’s difficult, so they need to hire an expert.
But, the reality is that the PID calculations
are simple arithmetic procedures. By con-
trast, an expert’s focus on the mathematics
distracts those intellects from the impor-
tant aspects of control, such as structuring
ratio, cascade and override, or choosing
appropriate modifications, anti-windup and
initialization procedures. The real experts
are not necessarily the mathematicians of
control theory. They are the ones who can
implement control.
There are many modifications to the PID
equation. Reset feedback, for example, is
an alternate method to incrementally adjust
the bias, which prevents integral windup,
and is especially useful in override and con-
straint strategies. Another common modifi-
cation is the rate-before-reset or interacting
controller, which can be created if incre-
mental changes to the bias are also based
on the anticipated error. For a discussion
of such modifications, see Rhinehart, R. R.,
H. L. Wade, and F. G. Shinskey in the Instru-
ment Engineers’ Handbook, Vol II, Process
Control and Analysis, 4th Edition, B. Liptak,
Editor, Section 2.3, “Control Modes – PID
Variations,” pp. 124-129, Taylor and Francis,
CRC Press, Boca Raton, Fla., 2005.
For a procedure to tune the controller,
see “Criteria and procedure for control-
ler tuning” by R.R. Rhinehart (Control, Jan
’17, p. 54-55, www.controlglobal.com/
articles/2017/criteria-and-procedure-for-
controller-tuning).
R. Russell Rhinehart, engineering coach, R3eda, North
Carolina State University, can be reached at russ@
r3eda.com.
The real experts are not necessarily
the mathematicians of control theory.
They are the ones who can implement control.
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eHandbook: Level Measurement, Spring 2019 20
Is global warming like level control?Comparing the processes sheds light on the problem of regulating Earth’s temperature.by Béla Lipták
QIn a previous column, you wrote that understanding the control of global warming is very similar to understanding level control. Could you
explain this in more detail? In what respect
are the two processes similar?
Z. Friedmann
A: You hit upon a large subject, one that will
take up a full chapter in my upcoming book
to be published by ISA. I will try to give you
a brief answer.
Figure 1 shows the rise of the surface tem-
perature of the oceans during the indus-
trial age. Now, if we look at the protection
against climate change as a “control loop,”
the measurement of that loop is that tem-
perature, which has increased by only about
1.5 °C during the past century, and today
we’re just beginning to see its consequenc-
es (melting ice, wildfires, hurricanes). This
rate of rise is still slow (2.0-3.0 °C per cen-
tury) but is accelerating. My estimate is it
TEMPERATURE OVER TIMEFigure 1: The average surface temperature of the world’s oceans, using the baseline of 1971 to 2000 average. The shaded band shows the range of uncertainty in the data based on the number of measurements collected and the precision of the measurements used. Source: EPA
Tem
pera
ture
ano
mal
y (°F
)
2.0
1.5
1.0
0.5
0
-0.5
-1.0
-1.5
-2.01880 1900 1920 1940 1960 1980 2000 2020
Year
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eHandbook: Level Measurement, Spring 2019 21
will reach a rise of about 5.0 °C by 2075. In
my view, we’ll never stop at 2.0°C (recom-
mended by the Paris Agreement of 2015)
and particularly not at 1.5 °C (recommended
by the U.N. Intergovernmental Panel on Cli-
mate Change – IPCC in 2018) because this
high-inertia process is totally out of control.
Some believe that because the numbers de-
scribing the temperature rise are small, the
problem we face is also small. This is not the
case. Let me compare them with our own
body temperature, which is accurately con-
trolled by our brain. The body temperature
of a healthy, resting adult human being is
98.6 °F (37.0 °C). Our “thermostat” (called
the hypothalamus, a portion of the brain)
controls body temperature. The span of our
thermostat is 36.4–37.1 °C (97.5–98.8 °F)
or about 0.7 °C. This thermostat turns on
shivering at 97.5 °F and initiates sweating at
98.8 °F. I mention this only to illustrate that
certain processes must be controlled within
small limits because small temperature
changes can have large effects.
According to all scientific data (www.nasa.
gov/topics/earth/features/co2-temperature.
html), the thermostat of global temperature
control is CO2 concentration in the atmo-
sphere. If it rises, the global temperature rises
(because the thermal insulation of the planet
increases), and when it dops, the planet cools.
During the past 1 million years, nature
“controlled” this concentration by keep-
ing the inflow of CO2 into the atmosphere
(generated by animal life and man) rough-
ly equal the outflow (intake of plants and
dissipation by the oceans), and therefore
the atmospheric concentration of CO2
stayed roughly constant, never exceeding
280 ppm even during ice ages, chang-
ing sun spot numbers, volcanic activity or
meteor impacts.
If we look at the atmosphere as a tank and
CO2 concentration as the level in that tank,
then we could say that this level stayed rea-
sonably constant for a million years because
it never exceeded 280 ppm (the planet didn’t
need to start “sweating”) as nature took care
of it. Since the beginning of the industrial
revolution, humans gradually took over this
control from nature, CO2 concentration in the
atmosphere increased from 280 to more than
410 ppm, and it’s predicted by most models
that it will reach or even exceed 500 ppm by
the end of this century.
If a control engineer was to bring this
process under control by returning CO2
concentration to the stable, pre-industrial
state, the task would be to balance the in
and outflows of this tank, and on top of
that, remove roughly half of the CO2 that
accumulated during the industrial age. If a
conventional level controller was installed
on this tank, it would see an error of 410 –
280 = 130 ppm and a past error accumula-
tion of some 400-500 Gt (vvvgigatons) of
carbon. It would immediately close the inlet
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eHandbook: Level Measurement, Spring 2019 22
valve and open the outlet valve.
Unfortunately, in this process, these valves
are stuck. The outflow from the tank (the
CO2 intake of the plants and dissipation
by the oceans) can’t be increased. In fact,
it has probably decreased during the past
century because of deforestation, acidifica-
tion of the oceans, and building of dams/
reservoirs, holding 8,000 km3 of water,
which also emit carbon to the atmosphere.
In short, this outlet valve is almost com-
pletely stuck and we have no technology
to open it further except reforestation,
which is unlikely due to overpopulation
(during the industrial age, population in-
creased from 1.0 to 9.0 billion).
As shown in Figure 2, every year we send 9
Gt of carbon into the atmosphere. 3 Gt of
that is taken up by the photosynthesis of
CARBON IMBALANCEFigure 2: This figure shows the fast carbon cycle (left - on land, right - in the oceans) in billions of tons of carbon per year. Yellow numbers are natural fluxes, red are human contributions. White numbers refer to stored carbon. Source: NASA
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eHandbook: Level Measurement, Spring 2019 23
plants, 2 Gt is dissipated by the oceans, and
4 Gt remains in the atmosphere for the next
20 to 200 years. So, the inflow exceeds the
outflow by 4 Gt/yr.
We do have some means to reduce this
inflow, such as using more bicycles, public
transport, converting to electric cars, insu-
lating our homes, using smart thermostats
and appliances, eliminating animal prod-
ucts from our diet (which cuts greenhouse
emissions by more than 10%), introduc-
ing carbon taxes (not cap-and-trade, but
taxes), and eventually, fully converting the
energy economy from fossil/nuclear fuels
to carbon-free ones. The speed of conver-
sion is a function of both the marketplace
and government support. Where both are
present, the conversion is faster (in Cali-
fornia today, green electricity is 30% of
the total), while where only the market-
place is driving the conversion, it is much
slower (15% in the U.S. overall).
It will probably take a generation or two
to overcome the resistance of the fossil
industry. Leaving some $35 trillion worth
of fossil fuel in the ground justifies some
resistance, and it will take time for society
as a whole to realize that we must con-
vert to solar energy and use hydrogen as
the means of storing, transporting and
distributing this energy to areas where
insolation is insufficient. (For more solar
storage technology, see my book, Post
Oil Energy Economy, or tune in the video:
http://techchannel.att.com/play-video.
cfm/2011/8/25/Science-&-Technology-
Author-Series-Bela-G-Liptak:-Post-Oil-
Energy-Technology.)
What even the best of our leaders seem
to not understand is that, even after we’ve
balanced the in and outflows, we will not
have returned the planet to pre-industrial
conditions because even after this tre-
mendous technological and political trans-
formation effort, we didn’t even start to
remove the already accumulated 400-500
Gt of carbon from the atmosphere, which
can stay there for 20-200 years. And, as
long as that accumulation remains, the
CO2 concentration does not drop and the
planet will keep warming.
This is obvious to a process control engi-
neer, but not to our well-intentioned leaders
who wrote the Paris Agreement in 2015 or
the smarter U.N. experts who participated
in the IPCC meeting in 2018. It is for this
reason that we who understand process
control have the responsibility to explain
that the present uncontrolled rate of carbon
accumulation will reach approximately 500
ppm by the end of this century, at which
point the tropical regions of the planet are
likely to become unlivable, and the resulting
biblical-scale migration could destroy hu-
man civilization.
Béla Lipták